In this contribution, we follow the spirit of the arguments brought in [1]} and adapt it to consider a flat boundary which is an important case in applications. The technique is interesting since it can be used for efficient numerical implementation of the equation using spectral methods based on the explicit formulas of Proposition REF . After discussing the generalities of the problem in the remainder of Section 1, we proceed to prove the central result of the paper about hypo-elliptic averaging for the equation with flat and prescribed boundary in Section 2. In Section 3, a priori estimates which lead to interior smoothness of solutions are established. Finally, in Section 4 with develop the well posedness theory of the model. The argument is fairly simple and based on the classical literature of the radiative transfer in convex domains.
